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One possible method of systematically counting the rectangles it to start by looking at all of the points that could be the top left corner of a rectangle. These have been numbered working from top left to bottom right.
Now take each of these 'corners' in turn and count how many rectangles can be found containing this corner at the top left. The results are shown in the table below:
Corner  Number of rectangles  View rectangles 
1  1  
2  3  
3  2  
4  3  
5  1  
6  5  
7  3  
8  2  
9  2  
10  2  
11  1  
12  1  
Total:  26 
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Have you read Craig's book yet?Craig Barton must surely be the voice of Mathematics teachers in the UK. His wonderful podcasts interviewing the industry experts have culminated in this wonderful book. As Craig says: "I genuinely believe I have never taught mathematics better, and my students have never learned more. I just wish I had known all of this twelve years ago..." more... "How I wish I'd taught Maths" is an extraordinary and important book. Part guide to research, part memoir, part survival handbook, it’s a wonderfully accessible guide to the latest research on teaching mathematics, presented in a disarmingly honest and readable way. I know of no other book that presents as much usable research evidence on the dos and don’ts of mathematics teaching in such a clear and practical way. No matter how long you have been doing it, if you teach mathematics—from primary school to university—this book is for you." Dylan Wiliam, Emeritus Professor of Educational Assessment, UCL. 
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Teacher, do your students have
access to computers? 

Here a concise URL for a version of this page without the comments.
Transum.org/go/?Start=November16
Here is the URL which will take them to another systematic listing activity.
Other shape counting starters:
How Many Squares 1?
 How Many Squares 2?
How Many Triangles 1?

How Many Triangles 2?  How Many Triangles 3?
How Many Rectangles?
 Rectangles Investigation 
Icosahedron 
Mystic Rose